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100+ Free MTEL Mathematics (63) Practice Questions

Pass your MTEL Mathematics Test (Field 63) exam on the first try — instant access, no signup required.

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According to the fundamental theorem of arithmetic, which statement is true about positive integers greater than 1?

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B
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D
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Key Facts: MTEL Mathematics (63) Exam

240

Passing Scaled Score

MTEL field 63 test information

85 MC + 2 OR

Test Format

MTEL field 63 overview

30%

Relations, Functions, and Algebra Weight

MTEL field 63 objectives

20%

Open-Response Subarea Weight

MTEL field 63 objectives

4 hours

Total Testing Time

MTEL field 63 overview

$139

Test Fee in 2026

Pearson MTEL registration

10%

Number Sense and Operations Weight

MTEL field 63 objectives

8-12

License Grade Span

Massachusetts DESE licensure

MTEL Mathematics (63) is Massachusetts' content-knowledge test for secondary math licensure. The current computer-based format has 85 multiple-choice questions plus 2 open-response items over 4 hours, with a 240 passing standard. Multiple-choice questions are distributed across four subareas: Number Sense and Operations (10%), Relations, Functions, and Algebra (30%), Geometry and Measurement (20%), and Probability, Statistics, Calculus, and Discrete Mathematics (20%); the two open-response items make up the Integration of Knowledge and Understanding subarea (20%). This free 100-question bank mirrors that blueprint so candidates can practice the heaviest-weight algebra and function content first, then close gaps in geometry, calculus, statistics, and discrete mathematics.

Sample MTEL Mathematics (63) Practice Questions

Try these sample questions to test your MTEL Mathematics (63) exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1What is the greatest common factor (GCF) of 84 and 126?
A.42
B.21
C.14
D.63
Explanation: Prime factorize: 84 = 2^2 x 3 x 7 and 126 = 2 x 3^2 x 7. The GCF takes the lowest power of each shared prime: 2^1 x 3^1 x 7^1 = 42.
2Which property of the real numbers is illustrated by the statement a(b + c) = ab + ac?
A.Distributive property
B.Associative property of addition
C.Commutative property of multiplication
D.Closure property
Explanation: The distributive property links multiplication over addition, stating that multiplying a sum by a factor equals the sum of the individual products. This is exactly the form a(b + c) = ab + ac.
3Expressed in scientific notation, what is the product (3.0 x 10^5)(4.0 x 10^-2)?
A.1.2 x 10^4
B.1.2 x 10^3
C.12 x 10^3
D.1.2 x 10^7
Explanation: Multiply the coefficients: 3.0 x 4.0 = 12. Add the exponents: 10^5 x 10^-2 = 10^3. So 12 x 10^3 = 1.2 x 10^4 in proper scientific notation.
4What is the value of the complex product (2 + 3i)(2 - 3i)?
A.13
B.-5
C.4 - 9i
D.13i
Explanation: This is a product of complex conjugates, which equals a^2 + b^2 for (a + bi)(a - bi). Here a = 2 and b = 3, so 2^2 + 3^2 = 4 + 9 = 13, a real number.
5Given matrices A = [[1, 2],[3, 4]] and B = [[0, 1],[1, 0]], what is the product AB?
A.[[2, 1],[4, 3]]
B.[[1, 2],[3, 4]]
C.[[0, 2],[3, 0]]
D.[[3, 4],[1, 2]]
Explanation: Multiplying by B (a permutation matrix that swaps columns) gives AB with columns of A interchanged. Row 1: [1x0+2x1, 1x1+2x0] = [2, 1]; Row 2: [3x0+4x1, 3x1+4x0] = [4, 3].
6What is the determinant of the matrix [[5, 2],[3, 4]]?
A.14
B.26
C.11
D.-14
Explanation: For a 2x2 matrix [[a, b],[c, d]], the determinant is ad - bc. Here that is (5)(4) - (2)(3) = 20 - 6 = 14.
7A student claims that the set of integers is closed under division. Which counterexample best disproves this claim?
A.7 divided by 2 equals 3.5, which is not an integer
B.6 divided by 3 equals 2, which is an integer
C.-8 divided by 4 equals -2, which is an integer
D.0 divided by 5 equals 0, which is an integer
Explanation: A set is closed under an operation only if the operation on any members always yields a member of the set. The example 7 / 2 = 3.5 produces a non-integer, proving integers are not closed under division.
8Using the Euclidean algorithm, what is the GCF of 1071 and 462?
A.21
B.42
C.7
D.33
Explanation: Apply the Euclidean algorithm: 1071 = 2(462) + 147; 462 = 3(147) + 21; 147 = 7(21) + 0. The last nonzero remainder, 21, is the GCF.
9What is the absolute value of the complex number 3 - 4i?
A.5
B.7
C.1
D.25
Explanation: The absolute value (modulus) of a + bi is the square root of a^2 + b^2. Here that is the square root of 3^2 + (-4)^2 = square root of 9 + 16 = square root of 25 = 5.
10What is the least common multiple (LCM) of 12 and 18?
A.36
B.72
C.6
D.216
Explanation: Prime factorize: 12 = 2^2 x 3 and 18 = 2 x 3^2. The LCM takes the highest power of each prime: 2^2 x 3^2 = 4 x 9 = 36.

About the MTEL Mathematics (63) Exam

The MTEL Mathematics test (Field 63) is the subject-matter exam required for the Massachusetts secondary mathematics (grades 8-12) educator license. It assesses content knowledge at the level of a baccalaureate mathematics major across five subareas, from number sense through calculus, statistics, and the integration of mathematical reasoning.

Questions

100 scored questions

Time Limit

4 hours

Passing Score

240 scaled score

Exam Fee

$139 per attempt (Massachusetts DESE / Pearson)

MTEL Mathematics (63) Exam Content Outline

10% of this bank

Number Sense and Operations

Mirrors Subarea I (10%): real number system properties, operations with integers, fractions, decimals, percents, exponents and scientific notation, number theory (primes, GCF, LCM, Euclidean algorithm), complex numbers, matrices, determinants, and vectors.

30% of this bank

Relations, Functions, and Algebra

Mirrors Subarea II (30%): algebraic techniques and equation solving, relations and functions, linear/quadratic/exponential/logarithmic/polynomial/rational/radical functions, sequences and series, systems, and trigonometric functions and identities.

20% of this bank

Geometry and Measurement

Mirrors Subarea III (20%): measurement and dimensional analysis, scaling effects, Euclidean axiomatic structure and proof, triangle and circle properties, 2D and 3D figures, coordinate and transformational geometry, conic sections, and non-Euclidean geometry.

40% of this bank

Probability, Statistics, Calculus, Discrete Math, and Integration

Mirrors Subareas IV (20%) and V (20%): probability and expected value, descriptive statistics and sampling, limits, derivatives, integrals, counting, set theory, logic, and induction, plus integration items that model and justify across at least two content areas.

How to Pass the MTEL Mathematics (63) Exam

What You Need to Know

  • Passing score: 240 scaled score
  • Exam length: 100 questions
  • Time limit: 4 hours
  • Exam fee: $139 per attempt

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

MTEL Mathematics (63) Study Tips from Top Performers

1Prioritize Relations, Functions, and Algebra because it is 30% of the multiple-choice score and underpins many other items
2Work the calculus objectives (limits, derivatives, integrals) since they are tested even though many candidates last studied them in college
3Use the official field 63 objectives as a checklist and rate your confidence on each before building a study schedule
4For geometry, practice both coordinate-based proofs and classical Euclidean reasoning, including constructions and conic sections
5Rehearse the open-response format by modeling real situations, labeling graphs, and writing complete justifications under time pressure
6Take mixed timed sets to build four-hour stamina, then review every miss by subarea to target your weakest content

Frequently Asked Questions

What is on the MTEL Mathematics (63) test?

Field 63 covers five subareas: Number Sense and Operations, Relations/Functions/Algebra, Geometry and Measurement, Probability/Statistics/Calculus/Discrete Mathematics, and an Integration of Knowledge subarea. The first four are tested with multiple-choice items, and the fifth is assessed by two open-response items.

How many questions are on the MTEL Mathematics (63) exam?

The current computer-based test has 85 multiple-choice questions plus 2 open-response items. The multiple-choice questions account for 80% of the score and the two open-response items account for 20%.

What is the passing score for MTEL Mathematics (63)?

The passing score is a scaled score of 240, consistent with most MTEL tests. You must meet that standard on this subject test in addition to any other tests required for your license.

How much does the MTEL Mathematics (63) test cost in 2026?

The Mathematics (63) test fee is $139 per attempt. Confirm the current amount in your Pearson registration account before checkout, since fees and bundle options can change.

Which content area is weighted most heavily on field 63?

Relations, Functions, and Algebra is the largest multiple-choice subarea at 30%, with roughly 31-33 questions. Geometry and Measurement and the combined Probability/Statistics/Calculus/Discrete subarea each contribute 20%, while Number Sense and Operations contributes 10%.

How should I prepare for the MTEL Mathematics (63) open-response items?

The two open-response items require you to integrate at least two content areas by building models, drawing labeled graphs or diagrams, applying mathematical techniques, and justifying conclusions. Practice writing organized, fully explained solutions rather than just final answers.